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Physical Review B

Long-range magnetic response of the XY spin chain under far-from-equilibrium conditions

Anna Gorczyca–Goraj, Marcin Mierzejewski, Tomaž Prosen

Published May 27, 2010• 5 citations
Publisher

Abstract

Within the formalism of the Keldysh Green's functions we investigate long-range response of an anisotropic $XY$ chain to the local magnetic field. This field couples to a single spin on a selected lattice site. The system is driven out of equilibrium by a coupling to two semi-infinite $XX$ spin chains. We demonstrate that the long-range response becomes enhanced by a few orders of magnitude upon application of nonequilibrium conditions. This enhancement does not occur in the isotropic $XX$ chain. Our results agree with the recently predicted nonequilibrium-driven long-range magnetic correlations [T. Prosen and I. Pi\ifmmode \check{z}\else \v{z}\fi{}orn, Phys. Rev. Lett. 101, 105701 (2008)]. We argue that this effect may be observed in quasi-one-dimensional triplet superconductors.

DOI
10.1103/physrevb.81.174430

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Within the formalism of the Keldysh Green's functions we investigate long-range response of an anisotropic XYXYXY chain to the local magnetic field. This field couples to a single spin on a selected lattice site. The system is driven out of equilibrium by a coupling to two semi-infinite XXXXXX spin chains. We demonstrate that the long-range response becomes enhanced by a few orders of magnitude upon application of nonequilibrium conditions. This enhancement does not occur in the isotropic XXXXXX chain. Our results agree with the recently predicted nonequilibrium-driven long-range magnetic correlations [T. Prosen and I. Pi\ifmmode \check{z}\else \v{z}\fi{}orn, Phys. Rev. Lett. 101, 105701 (2008)]. We argue that this effect may be observed in quasi-one-dimensional triplet superconductors.
Out-of-equilibrium behavior is explored in the one-dimensional anisotropic XY model. Initially preparing the system in the isotropic XX model with a linearly varying magnetic field to create a domain-wall magnetization profile, dynamics is generated by rapidly changing the exchange interaction anisotropy and external magnetic field. Relaxation to a nonequilibrium steady state is studied analytically at the critical transverse Ising point, where correlation functions may be computed in closed form. For arbitrary values of anisotropy and external field, an effective generalized Gibbs' ensemble is shown to accurately describe observables in the long-time limit. Additionally, we find spatial oscillations in the exponentially decaying, transverse spin-spin correlation functions with wavelength set by the magnetization jump across the initial domain wall. This wavelength depends only weakly on anisotropy and magnetic field in contrast to the current, which is highly dependent on these parameters.
We conjecture that nonequilibrium boundary conditions generically trigger long-range order in nonequilibrium steady states of locally interacting quantum chains. Our result is based on large scale density matrix renormalization group simulations of several models of quantum spin-1/2 chains which are driven far from equilibrium by coupling to a pair of unequal Lindblad reservoirs attached locally to the ends of the chain. In particular, we find a phase transition from exponentially decaying to long-range spin-spin correlations in an integrable Heisenberg XXZ chain by changing the anisotropy parameter. Long-range order also typically emerges after breaking the integrability of the model.
Explicit solution for the two-point correlation function in a non-equilibrium steady state of a nearly isotropic boundary driven open XY spin 1/2 chain in the Lindblad formulation is provided. A non-equilibrium quantum phase transition from exponentially decaying correlations to long range order is discussed analytically. In the regime of long range order a new phenomenon of correlation resonances is reported, where the correlation response of the system is unusually high for certain discrete values of the external bulk parameter, e.g. the magnetic field.
Explicit solution for the 2-point correlation function in a non-equilibrium steady state of a nearly isotropic boundary-driven open XY spin 1/2 chain in the Lindblad formulation is provided. A non-equilibrium quantum phase transition from exponentially decaying correlations to long-range order is discussed analytically. In the regime of long-range order a new phenomenon of correlation resonances is reported, where the correlation response of the system is unusually high for certain discrete values of the external bulk parameter, e.g. the magnetic field.
We study the equilibrium properties of the spin-1/21/21/2 XY chain with an infinite-range transverse interaction. At zero temperature, competition between the XY- and zzz-ordered phases induced by the infinite-range interactions gives rise to a first-order transition upon increasing the transverse coupling. We show that the two gapless points of the XY model behave in fundamentally different ways: isotropic spin chains experience a first-order transition at finite coupling; maximal anisotropic chains overcome a nonalgebraic phase transition at zero coupling strength. The phase diagram depicts a first-order reentrant transition that turns second order along a tricritical line separating a paramagnetic phase from an ordered one at finite temperature. The mean-field approach captures the local properties of the eigenstates and reveals the appearance of a magnetization gap in the spectrum. Global properties, e.g., entanglement entropy, are well approximated only at spectral boundaries. The mean entanglement entropy and the level-spacing ratio deviate from the Gaussian results, revealing the interacting nature of the problem.
We investigate the Heisenberg XXZXXZXXZ chain with long-range interactions in the ZZZ dimension. By applying two magnetic boundary reservoirs, we drive the system out of equilibrium and induce a nonzero steady-state current. The long-range coupled chain shows nearly ballistic transport and linear response for all potential differences of the external reservoirs. In contrast, the common isotropic nearest-neighbor coupling shows negative differential conductivity and a transition from diffusive to subdiffusive transport for a far-from-equilibrium driving. Adding disorder, the change in the transport for nearest-neighbor coupling is therefore highly dependent on the driving. We find for the disordered long-range coupled XXZXXZXXZ chain, any change in the transport behavior is independent of the potential difference and the coupling strengths of the external reservoirs.